Shift and subtract

The idea

A pure repeating decimal repeats immediately after the decimal point; to write it as p/q, let x be the decimal, multiply by 10ⁿ where n is the number of repeating digits, subtract x, and solve.

A decimal that never stops

A decimal that stops is easy to write as a fraction: 0.35 is 35/100, which is 7/20. But what about 0.666…, where the 6 goes on forever?

Which fraction is equal to 0.666…?

• 6/10
• 2/3
• 66/100

Line up the tails

Let x be the decimal. Shift the point to the right by choosing how many places. When the repeating tails line up, subtract: the tails cancel.

10^n x − x

A pure repeating decimal repeats immediately after the decimal point. To write it as p/q, let x be the decimal, multiply by 10^n where n is the number of repeating digits, subtract x, and solve.

For x = 0.666…: 10x = 6.666…, so 10x − x = 6 and 9x = 6. Then x is 6/9, which is 2/3.

Notes

Pure repeating decimal: let x be the decimal, multiply by 10^n (n = number of repeating digits), subtract x, and solve for x.

Check yourself

x = 0.777…, so 10x = 7.777…. Subtract: 10x − x = 9x = ?

Answer: 7

7.777… − 0.777… = 7, so 9x = 7 and x = 7/9.

x = 0.1212… (the two digits 12 repeat). By what number do we multiply x?

Write 0.333… as p/q. Then 10x − x = 3, so 9x = 3. In lowest terms, what is the denominator?

Answer: 3

x = 3/9 = 1/3, so the denominator is 3.

Why do we subtract x from 10^n x?

• 10. Only one place would not line up the tails. Two digits repeat.
• 12. We multiply by a power of 10, not by the repeating digits.
• 100 — correct. Yes. Two digits repeat, so we multiply by 10² = 100.
• To make the number smaller. The aim is to get rid of the endless tail.
• The repeating tails cancel, leaving a whole number — correct. Yes. Both numbers have the same endless tail, so the difference is a whole number.
• Because the rule says so. There is a reason: the tails cancel.